Let \(N\)  be a prime number and let \(S_2(\Gamma _0(N),\mathbb {Z})\)  denote the modular forms of weight 2 and level \(\Gamma _0(N)\)  with integer coefficients, and for any other ring \(R\) , we denote \(S_2(\Gamma _0(N),R)=S_2(\Gamma _0(N),\mathbb {Z})\otimes R\) . If \(R\)  is a characteristic \(p\)  ring, we define \(S_2(\Gamma _0(N),R)^{\text{Katz}}\)  to be the \(R\) -module of Katz forms as defined in [1]}, and define similar notation for the spaces of weight 1 forms. For \(N\nmid n\) , let \(T_n\)  denote the \(n\) th Hecke operator inside \(\operatorname{End}(S_2(\Gamma _0(N),\overline{\mathbb {Z}}))\) , and let \(U_N\)  denote the \(N\) th Hecke operator. We let \({\operatorname{an}}\)  denote \(\mathbb {Z}[T_3, T_5, \ldots ]\) , the algebra generated by \(T_n\)  for \((2N, n)=1\) , and we denote \({\operatorname{an}}[T_2, U_N]\)  by \(. The goal of this paper is to compute the index of \) an\( inside \) . Specifically, we prove the following theorem in sections  and :
